#### Volume 22, issue 2 (2018)

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Long-time behavior of $3$–dimensional Ricci flow, D: Proof of the main results

### Richard H Bamler

Geometry & Topology 22 (2018) 949–1068
##### Abstract

This is the fourth and last part of a series of papers on the long-time behavior of $3$–dimensional Ricci flows with surgery. In this paper, we prove our main two results. The first result states that if the surgeries are performed correctly, then the flow becomes nonsingular eventually and the curvature is bounded by $C{t}^{-1}$. The second result provides a qualitative description of the geometry as $t\to \infty$.

##### Keywords
Ricci flow, Ricci flow with surgery, finitely many surgeries, asymptotics of Ricci flow, collapsing theory of $3$–manifolds, topology of $3$–manifolds, geometrization conjecture
##### Mathematical Subject Classification 2010
Primary: 53C44
Secondary: 49Q05, 53C23, 57M15, 57M20
##### Publication
Revised: 22 December 2016
Accepted: 21 January 2017
Published: 16 January 2018
Proposed: Tobias H. Colding
Seconded: Bruce Kleiner, Gang Tian
##### Authors
 Richard H Bamler Department of Mathematics University of California Berkeley, CA United States https://math.berkeley.edu/~rbamler/